Tits alternative for automorphism groups of compact Kähler manifolds
This survey focuses on some properties of the automorphisms groups of compact Kähler manifolds. In particular, the main result proved is a result by \textit{J. Keum, K. Oguiso} and \textit{D.-Q. Zhang}, Math. Res. Lett. 16, No. 1, 133--148 (2009; Zbl 1172.14025)] and by \textit{D.-Q Zhang} [Invent. Math. 176, No. 3, 449--459 (2009; Zbl 1170.14029)]. Results by \textit{S. Cantat} [Ann. Math. (2) 174, No. 1, 299--340 (2011; Zbl 1233.14011)] and by Campana-Wang-Zhang are also reviewed. Let \(X\) be a compact Kähler manifold of dimension \(k\) and of Kodaira dimension \(\kappa_X\); take \(\kappa:=\max (\kappa_X,0)\) if \(\kappa_X>k\), \(\kappa:=k-1\) otherwise. Let \(G\) be a group of holomorphic automorphisms of \(X\) which does not contain any free non-abelian subgroup. Then it is shown that \(G\) admits a finite index subgroup \(G'\) such that: (i) \(G'\) is solvable, or equivalently, \(\mathrm{Aut}(X)\) satisfies the Tits alternative; (ii) the set \(N'\) of zero entropy elements of \(G'\) is a normal subgroup of \(G'\) and \(G'/N'\) is a free abelian group of rank at most \(k-\kappa-1\). All ideas necessary to the proof are reviewed in the first sections of the paper, avoiding technicalities: a mixed version of the Hodge-Riemann theorem, topological entropy and dynamical degrees and finally fibrations and the notion of relative dynamical degree. The last section contains the details of the proof of the main theorem cited above.
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